obtained by Means of imaginary Quantities. 117 
with imaginary symbols are intelligible and just, the only argu- 
ment for their exclusion must be founded on the existence of 
methods more general and expeditious. 
The application of imaginary quantities to the theory of equa- 
tions, has perhaps been made more extensively than to any other 
part of analysis. To consider the propriety of this application 
on the grounds of perspicuity and conciseness, a long discussion 
would be necessary. I may, however, be here permitted 
merely to state my opinion, that impossible quantities must be 
employed in the theory of equations, in order to obtain general 
rules and compendious methods. The demonstration of the 
principal proposition, that every root of an equation is com- 
prised under the form M -f N V — 1, and that consequently 
every equation of 2 n dimensions, is always divisible into n qua- 
dratic factors, appears to me, I confess, deficient in evidence 
and mathematical rigour. To establish this proposition, and to 
prove likewise, that every imaginary expression derived from 
transcendental operations is always comprised under the form 
M -f N \/ — 1, is the object of two Memoirs by D’Alembert 
and Euler. (Mem. de Berlin, 1746, 1749.) 
M. Foncenex, (Mem.de Turin.) Lagrange, (Mem.de Berlin, 
1771, 1772, 1773,) Laplace, Waring, and other mathema- 
ticians, have directed their inquiries towards the same subject. * 
values are necessarily and by strict consequence true : and not true because they may 
be verified by a distinct or more rigorous investigation, nor because the operations 
have a tacit and implied reference to other more legitimate operations. 
* None of the demonstrations go farther than to shew the possibility of resolving 
an equation of 2 n dimensions into n quadratic factors. The actual resolution of equa- 
tions that pass the fourth degree, has not hitherto been executed. Of the labours of 
such learned men as those I have mentioned, I speak with the greatest diffidence; the 
mere knowledge, however, of the possibility of the resolution of equations, appears to 
